This might not really be completely maths related, but more physics related, but I still hope someone might push me in the right direction or maybe give me some articles about this problem.
I'm doing a project at school involving a directional sound speaker, or a speaker that emits sound only in one direction, instead of nearly omni-directional.
We have achieved this by using a large number of small piezo-speakers (let's say 8 for the sake of simplicity), positioned in a line with the distance between each speaker exactly half the length of the wavelength of the sound that the speakers produce.
Because all the speakers are driven by the same function generator, they are exactly in phase with eachother.
If you would 'look' at the phase-difference exactly in front of the line of speakers, you would notice that all soundwaves are exactly in phase with eachother, producing constructive interference (adding up the waves).
If you look at a right angle to the line of speakers (on the side), all the soundwaves are exactly out of phase with eachother, producing destructive interference ('cancelling out' the sound).
Speakers 1 and 2, speakers 3 and 4, speakers 5 and 6 and speakers 7 and 8 cancel eachother out.
We have tested this system with 52 speakers in a hexagonal pattern and it works pretty well.
Now the hard part.
Between 'right infront of' (angle = 0) and 'at a right angle' (angle = 90), interference occurs aswell, only at a different scale, since the waves will not cancel out exactly or add up exactly.
Now, we would like to have some theoritical stuff in our project, in detail, we would like to calculate beforehand how loud the speakers will sound in any given angle (between -90 and 90 degrees), and then measure it with our device and compare the results.
However, we have not yet managed to work out how to do this...
We have thought about it, searched on the internet but couldn't really find anything.
Our main idea's are that, since the speakers are all in phase, the phase difference between two speakers depends only on the distance from the speakers.
If you would take one point and call the distance from that point to speaker 1 r1' and the distance from that point to speaker 2 r2, then I'm absolutely sure you could work out the phase difference between the two speakers.
The phase difference would in turn give us some idea of how the soundwaves will interfere.
So to sum it up, can you give me some advice, tips, help on how to calculate the phase difference from two sources, depending on the distance from those sources? I'm sure we can work out for ourselves how it would go with multiple sources.
I quickly racked up an illustrative image, see attachment.
Points A and B are what I talked about; point A is at a right angle (90 degrees) to the sound sources and the waves from both sources will have a phase difference of exactly pi rad (or 180 degrees) which means they will cancel out.
In point B, the phase difference will be 0, since the distance to both sources is equal, and the waves will add up.
In point P however, it is not directly obvious how the waves will interfere.
I'm sure the calculation here might be very simple but we are not seeing it... Perhaps we're thinking too complicated, or perhaps too simple...
I would be grateful if anyone had anything useful to say on this, thanks.
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